Green's function coupled-cluster approach: Simulating photoelectron

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Spectroscopy and Excited States

Green's function coupled-cluster approach: Simulating photoelectron spectra for molecular systems Bo Peng, and Karol Kowalski J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00313 • Publication Date (Web): 29 Jun 2018 Downloaded from http://pubs.acs.org on July 1, 2018

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Journal of Chemical Theory and Computation

Green’s function coupled-cluster approach: Simulating photoelectron spectra for realistic molecular systems Bo Peng⇤ and Karol Kowalski⇤ William R. Wiley Environmental Molecular Sciences Laboratory, Battelle, Pacific Northwest National Laboratory, K8-91, P.O. Box 999, Richland, WA 99352, USA E-mail: [email protected]; [email protected]

∗

To whom correspondence should be addressed

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Abstract In this paper we present an efficient implementation for the analytical energydependent Green’s function coupled-cluster with singles and doubles (GFCCSD) approach with our first practice being computing spectral functions of realistic molecular systems. Due to its algebraic structure, the presented method is highly scalable, and is capable of computing spectral function for a given molecular system in any energy region. Several typical examples have been given to demonstrate its capability of computing spectral functions not only in the valence band, but also in the core-level energy region. Satellite peaks have been observed in the inner valence band and core-level energy region where many-body e↵ect becomes significant and single particle picture of ionization often breaks down. The accuracy test has been carried out by extensively comparing the computed spectral functions by our GFCCSD method with experimental photoelectron spectra as well as the theoretical ionization potentials obtained from other methods. It turns out GFCCSD method is able to provide qualitative or semiquantitative level of description of ionization processes in both the core and valence regimes. To significantly improve the GFCCSD results for the main ionic states, larger basis set can usually be employed, whereas the improvement of the GFCCSD results for the satellite states needs higher order many-body terms to be included in the GFCC implementation.

Introduction Since first introduced to chemical and condensed matter communities through electronic structure theory, 1–4 Green’s function based formalisms have been widely exploited in the past few decades to describe molecular electronic structure, compute beyond-ground-state electronic properties, and help interpret the spectra of molecular systems in many emerging physical and chemical areas including photovoltaics, semiconductor, magnetoresistance, and novel superconductivity to name just a few. 5–40 One conceptual merit of the one-particle

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many-body Green’s function (MBGF) is its direct way of calculating the key electronic properties of the ionization and attachment process (without resorting to separate calculations for di↵erent states). Applicable one-particle MBGF approaches can usually be derived from the many-body perturbation expansions for the matrix of one-particle Green’s function G and/or for the related self-energy ⌃ via the Dyson equation. Typical approaches include GW method, 5,27–31,41 outer-valence Green’s function (OVGF) method, 6–9 and algebraic-diagrammatic construction (ADC) 7,10,11 approximation scheme. In the GW method, the self-energy is calculated from the product of the G matrix (usually obtained from Hartree-Fock, HF, or density-functional theory, DFT) and the lowest order W term (the so-called screened Coulomb interaction). Higher orders in W could be added by iterating the equations, which will return the self-energy into the computation of G and achieve a self-consistency via the relationship ⌃ = iGW . In the OVGF method, the second and third order perturbation corrections of the self-energy are explicitly included, and the higher order contributions are approximated through a renormalization procedure. Both the GW method and OVGF method have been proved by numerous studies of weakly and moderately correlated molecular systems to provide accurate single particle properties (see, for example, Refs. 9,31,41–48). However, when many-body e↵ects become crucial, as often featured by the satellite states in the ionization process out of the inner valence band where poles will appear in the analytical structure of the self-energy, 49–51 a proper description of the poles in the analytical structures of the self-energy is required. Following this direction, the most successful approach is the ADC approximate scheme, especially the third-order ADC method using the Dyson equation or a more direct non-Dyson framework. The ADC scheme is based on the diagrammatic perturbation theory of the polarization propagator, and exploits an algebraic scheme to represent the infinite partial summations of the perturbation series of the self-energy exact up to a finite order. Since it was proposed, numerous applications have been reported for the core and valence band calculations of small- and medium-sized molecules (see Ref. 11 for a recent review).

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Alternatively, the description of the many-body e↵ect can be improved by incorporating the one-particle Green’s function with correlated wave function expansions. For example, the possibility of utilizing highly-correlated methodologies to describe local Green’s function or corresponding self-energies in dynamical mean field theories (DMFT) has drawn considerable interest recently. 52–58 From this perspective, coupled-cluster (CC) formalism 59–65 is one of the ideal candidates to provide hierarchical classes of CC approximations that are able to generate systematically improvable correlated wave function to efficiently capture various types of many-body e↵ects. The resulting Green’s function coupled-cluster (GFCC) formalism, since first introduced in 1990s (see Refs. 12–16, in particular the analytical and explicit GFCC formalism first proposed by Nooijen and Snijder, see Refs. 13–15), has drawn a lot of attention. For example, by realizing the equivalence between GFCC and equationof-motion coupled-cluster (EOMCC), the pole positions in the GFCC structure representing ionization potentials or electron affinities can be directly obtained by diagonalizing the ionization potential (or electron attachment) equation-of-motion coupled-cluster (IP/EAEOMCC) Hamiltonian matrix. However, to solve for roots that are embedded deeply inside the Hamiltonian matrix, more robust eigen-solvers 66,67 are required to skip the low-lying roots and save computational cost. E↵orts have also been made to compute the GFCC matrix elements directly. Recently, we reported the direct computation of the GFCCSD matrix elements for some small molecules at some frequencies values, 37,38 and slightly later, McClain et al. reported the computation of the spectral function of a uniform electron gas. 36 In analogy to Ref. 36, our previous work chose to first solve a set of linear equations for IP/EA-EOMCC type vectors, and then contracting these vectors with converged amplitudes of the CC ⇤ de-excitation operators to get the GFCCSD matrix elements. The analytical approach was designed to calculate GFCC for the whole complex plane, therefore includes all poles of the GFCC structure. Following this approach, we then were able to prove the connected character of the diagrams contributing to GFCC matrix elements, as well as the connected character of its n-th order derivative with respect to the energy and the

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corresponding CC self-energy operators, which provided a useful guidance for designing and analyzing new GFCC approximation schemes to include higher order correlation e↵ects. 39,40 In our first implementation of the GFCC approach employing single and double excitations in the cluster operator (GFCCSD) in the real space, we calculated the ionization potentials of some benchmark systems, which exhibited good agreement with the experimental observation. Moreover, we were also able to obtain the GFCCSD matrix elements and the corresponding self-energy for small molecules at a few frequency values. In this paper, in order to extend our approach to the whole complex plane and to calculate spectral functions of realistic molecular systems for any given energy regime, we propose an efficient implementation of our GFCCSD approach. The proposed implementation solves the real part and imaginary part of the above-mentioned linear equations in a coupled manner, which greatly saves computation cost in comparison with previous proposed implementation. Furthermore, our implementation is highly scalable, which enables the computation of GFCCSD matrix over the entire molecular orbital space for any given energy region in a relatively short period of time. Our accuracy tests focus on calculating spectral functions of six molecular systems of small and medium size, namely H2 O, N2 , CO, s-trans-1,3-butadiene, benzene, and adenine molecules, for which abundant theoretical and experimental results are available for extensive comparisons and discussions.

Methodology Our analytical GFCC expression is similar to the general GFCC formalism introduced by Nooijen et al.,, 13–15 and its specific formulations/properties and explicit solving procedure have been discussed in Refs. 37–40. By employing CC bi-variational approach, the corresponding energy-dependent Green’s function for an N -electron system can be expressed

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as T † aq (!

Gpq (!) = h |(1 + ⇤)e

T

h |(1 + ⇤)e

+ (H

ap (!

E0 )

i⌘) 1 ap eT | i +

E0 ) + i⌘) 1 a†q eT | i .

(H

(1)

Here, | i is the reference function (with energy E0 ), and the ap (a†p ) operator is the annihilation (creation) operator for electron in the p-th spin-orbital. The ! parameter denotes the energy (or frequency), and the imaginary part ⌘ is often called a broadening factor. The cluster operator T , defining correlated ground-state wave function | i for the N -electron system in CC parametrization | i = eT | i ,

(2)

and de-excitation operator ⇤ are defined as

T =

N X

X 1 ti1 ...in a† . . . a†an ain . . . ai1 , (n!)2 i ,...,i ; a1 ...an a1

(3)

N X

X 1 (n!)2 i ,...,i ;

(4)

n=1

⇤ =

n=1

n with tia11...i ...an and

a1 ...an i1 ...in

n 1 a1 ,...,an

1

a1 ...an † i1 ...in ai1

. . . a†in aan . . . aa1 ,

n

a1 ,...,an

being the antisymmetric amplitudes, and the indices i, j, k, . . . (i1 , i2 , . . .)

and a, b, c, . . . (a1 , a2 , . . .) corresponding to occupied and unoccupied spin-orbitals in the reference function | i respectively. representing the projection onto the subspace spanned by excited configurations |

a1 ...an i1 ...in i

defined as a†a1 . . . a†an ain . . . ai1 | i.

By applying the resolution of identity 1 = e

T T

e , the algebraic expression for matrix

elements of the Green’s function can be re-written as ¯ ¯N Gpq (!) = h |(1 + ⇤)a†q (! + H h |(1 + ⇤)¯ ap (!

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i⌘) 1 a ¯p | i +

¯ N + i⌘) 1 a H ¯†q | i ,


(5)

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¯ N (in normal product form representation), a where the similarity transformed operator H ¯p , ¯ and a†q are defined as ¯N = e H

T

H eT

E0 ,

(6)

a ¯p = e

T

ap eT = ap + [ap , T ] ,

(7)

¯ a†q = e

T † aq

eT = a†q + [a†q , T ] .

(8)

Now we can define an energy-dependent IP-EOMCC type operator Xp (!) X

Xp (!) =

xi (p, !)ai +

i

X

† xij a (p, !)aa aj ai + . . . ,

(9)

i

Green's function coupled-cluster approach: Simulating photoelectron

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